Solve 6m^2+420m+414

$$6 m ^ { 2 } + 420 m + 414$$

Factor

$$6\left(m+1\right)\left(m+69\right)$$

Solution Steps

Factor out 6.

$$6\left(m^{2}+70m+69\right)$$

Consider m^{2}+70m+69. Factor the expression by grouping. First, the expression needs to be rewritten as m^{2}+am+bm+69. To find a and b, set up a system to be solved.

$$a+b=70 ab=1\times 69=69$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 69.

$$1,69 3,23$$

Calculate the sum for each pair.

$$1+69=70 3+23=26$$

The solution is the pair that gives sum 70.

$$a=1 b=69$$

Rewrite m^{2}+70m+69 as \left(m^{2}+m\right)+\left(69m+69\right).

$$\left(m^{2}+m\right)+\left(69m+69\right)$$

Factor out m in the first and 69 in the second group.

$$m\left(m+1\right)+69\left(m+1\right)$$

Factor out common term m+1 by using distributive property.

$$\left(m+1\right)\left(m+69\right)$$

Rewrite the complete factored expression.

$$6\left(m+1\right)\left(m+69\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$6m^{2}+420m+414=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$m=\frac{-420±\sqrt{420^{2}-4\times 6\times 414}}{2\times 6}$$

Square 420.

$$m=\frac{-420±\sqrt{176400-4\times 6\times 414}}{2\times 6}$$

Multiply -4 times 6.

$$m=\frac{-420±\sqrt{176400-24\times 414}}{2\times 6}$$

Multiply -24 times 414.

$$m=\frac{-420±\sqrt{176400-9936}}{2\times 6}$$

Add 176400 to -9936.

$$m=\frac{-420±\sqrt{166464}}{2\times 6}$$

Take the square root of 166464.

$$m=\frac{-420±408}{2\times 6}$$

Multiply 2 times 6.

$$m=\frac{-420±408}{12}$$

Now solve the equation m=\frac{-420±408}{12} when ± is plus. Add -420 to 408.

$$m=-\frac{12}{12}$$

Divide -12 by 12.

$$m=-1$$

Now solve the equation m=\frac{-420±408}{12} when ± is minus. Subtract 408 from -420.

$$m=-\frac{828}{12}$$

Divide -828 by 12.

$$m=-69$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -1 for x_{1} and -69 for x_{2}.

$$6m^{2}+420m+414=6\left(m-\left(-1\right)\right)\left(m-\left(-69\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$6m^{2}+420m+414=6\left(m+1\right)\left(m+69\right)$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 6

$$x ^ 2 +70x +69 = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = -70 rs = 69$$

Two numbers r and s sum up to -70 exactly when the average of the two numbers is \frac{1}{2}*-70 = -35. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = -35 - u s = -35 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 69

$$(-35 - u) (-35 + u) = 69$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$1225 - u^2 = 69$$

Simplify the expression by subtracting 1225 on both sides

$$-u^2 = 69-1225 = -1156$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = 1156 u = \pm\sqrt{1156} = \pm 34$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =-35 - 34 = -69 s = -35 + 34 = -1$$

Evaluate

$$6\left(m+1\right)\left(m+69\right)$$